# Quantitative combinatorial geometry for concave functions

## TL;DR

This paper develops precise quantitative versions of Helly's and Tverberg's theorems, identifying conditions for large intersection witnesses under concave measures and analyzing approximation complexities.

## Contribution

It introduces new quantitative Helly-type theorems for convex set intersections with concave measures, including colorful and fractional variants, and studies approximation complexities.

## Key findings

- Characterization of conditions for large intersection witnesses
- Introduction of colorful and fractional Helly variants
- Bounds on approximation complexity for convex set families

## Abstract

We prove several exact quantitative versions of Helly's and Tverberg's theorems, which guarantee that a finite family of convex sets in $R^d$ has a large intersection. Our results characterize conditions that are sufficient for the intersection of a family of convex sets to contain a "witness set" which is large under some concave or log-concave measure. The possible witness sets include ellipsoids, zonotopes, and $H$-convex sets. Our results also bound the complexity of finding the best approximation of a family of convex sets by a single zonotope or by a single $H$-convex set. We obtain colorful and fractional variants of all our Helly-type theorems.